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Properties

Label 6-290e3-1.1-c1e3-0-1
Degree 66
Conductor 2438900024389000
Sign 11
Analytic cond. 12.417212.4172
Root an. cond. 1.521721.52172
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3·3-s + 6·4-s − 3·5-s + 9·6-s + 3·7-s + 10·8-s + 3·9-s − 9·10-s + 18·12-s + 3·13-s + 9·14-s − 9·15-s + 15·16-s + 3·17-s + 9·18-s − 18·20-s + 9·21-s − 15·23-s + 30·24-s + 6·25-s + 9·26-s + 27-s + 18·28-s − 3·29-s − 27·30-s − 9·31-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 3·4-s − 1.34·5-s + 3.67·6-s + 1.13·7-s + 3.53·8-s + 9-s − 2.84·10-s + 5.19·12-s + 0.832·13-s + 2.40·14-s − 2.32·15-s + 15/4·16-s + 0.727·17-s + 2.12·18-s − 4.02·20-s + 1.96·21-s − 3.12·23-s + 6.12·24-s + 6/5·25-s + 1.76·26-s + 0.192·27-s + 3.40·28-s − 0.557·29-s − 4.92·30-s − 1.61·31-s + ⋯

Functional equation

Λ(s)=(24389000s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 24389000 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(24389000s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 24389000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 66
Conductor: 2438900024389000    =    23⋅53⋅2932^{3} \cdot 5^{3} \cdot 29^{3}
Sign: 11
Analytic conductor: 12.417212.4172
Root analytic conductor: 1.521721.52172
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 24389000, ( :1/2,1/2,1/2), 1)(6,\ 24389000,\ (\ :1/2, 1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 10.1451654410.14516544
L(12)L(\frac12) ≈\approx 10.1451654410.14516544
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1−T)3 ( 1 - T )^{3}
5C1C_1 (1+T)3 ( 1 + T )^{3}
29C1C_1 (1+T)3 ( 1 + T )^{3}
good3S4×C2S_4\times C_2 1−pT+2pT2−10T3+2p2T4−p3T5+p3T6 1 - p T + 2 p T^{2} - 10 T^{3} + 2 p^{2} T^{4} - p^{3} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1−3T+6T2+4T3+6pT4−3p2T5+p3T6 1 - 3 T + 6 T^{2} + 4 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+9T2−24T3+9pT4+p3T6 1 + 9 T^{2} - 24 T^{3} + 9 p T^{4} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−3T+6T2+40T3+6pT4−3p2T5+p3T6 1 - 3 T + 6 T^{2} + 40 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−3T+42T2−84T3+42pT4−3p2T5+p3T6 1 - 3 T + 42 T^{2} - 84 T^{3} + 42 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1+9T2−56T3+9pT4+p3T6 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+15T+108T2+24pT3+108pT4+15p2T5+p3T6 1 + 15 T + 108 T^{2} + 24 p T^{3} + 108 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1+9T+108T2+556T3+108pT4+9p2T5+p3T6 1 + 9 T + 108 T^{2} + 556 T^{3} + 108 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+39T2+232T3+39pT4+p3T6 1 + 39 T^{2} + 232 T^{3} + 39 p T^{4} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+12T+99T2+528T3+99pT4+12p2T5+p3T6 1 + 12 T + 99 T^{2} + 528 T^{3} + 99 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−9T+144T2−758T3+144pT4−9p2T5+p3T6 1 - 9 T + 144 T^{2} - 758 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+45T2+192T3+45pT4+p3T6 1 + 45 T^{2} + 192 T^{3} + 45 p T^{4} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−9T+78T2−468T3+78pT4−9p2T5+p3T6 1 - 9 T + 78 T^{2} - 468 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+15T+180T2+1602T3+180pT4+15p2T5+p3T6 1 + 15 T + 180 T^{2} + 1602 T^{3} + 180 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−15T+204T2−1604T3+204pT4−15p2T5+p3T6 1 - 15 T + 204 T^{2} - 1604 T^{3} + 204 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−18T+3pT2−1676T3+3p2T4−18p2T5+p3T6 1 - 18 T + 3 p T^{2} - 1676 T^{3} + 3 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
71C2C_2 (1+12T+pT2)3 ( 1 + 12 T + p T^{2} )^{3}
73S4×C2S_4\times C_2 1+9T+234T2+1312T3+234pT4+9p2T5+p3T6 1 + 9 T + 234 T^{2} + 1312 T^{3} + 234 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−9T+102T2−320T3+102pT4−9p2T5+p3T6 1 - 9 T + 102 T^{2} - 320 T^{3} + 102 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−12T+3pT2−1920T3+3p2T4−12p2T5+p3T6 1 - 12 T + 3 p T^{2} - 1920 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+243T2−24T3+243pT4+p3T6 1 + 243 T^{2} - 24 T^{3} + 243 p T^{4} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−9T+282T2−1748T3+282pT4−9p2T5+p3T6 1 - 9 T + 282 T^{2} - 1748 T^{3} + 282 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.63558349084430931817762020080, −10.39576802160501566254891546059, −9.903128644917207584138040105758, −9.900102189037536620764853225375, −8.807995506409440880056456620375, −8.780491499792404110645782353211, −8.740768763493239935399568839108, −7.965254244668962494130985274267, −7.897885815155730349389946441766, −7.75234000453503620370712933227, −7.27335930412895876702869607926, −7.08820154662815520981181126867, −6.43990451466365518223286679974, −5.88306303986387319098906987374, −5.83393683358686715852853527128, −5.38584246802692058432397017240, −4.67807927963898269299364813226, −4.58550918474849529536313996909, −3.99536896728196627920557218645, −3.75702713742074221972192946275, −3.40347079805317736636773815143, −3.26772954396411075380503340706, −2.35928655279984995783914105728, −2.11697422565417834337721711706, −1.52241067737148877215906531093, 1.52241067737148877215906531093, 2.11697422565417834337721711706, 2.35928655279984995783914105728, 3.26772954396411075380503340706, 3.40347079805317736636773815143, 3.75702713742074221972192946275, 3.99536896728196627920557218645, 4.58550918474849529536313996909, 4.67807927963898269299364813226, 5.38584246802692058432397017240, 5.83393683358686715852853527128, 5.88306303986387319098906987374, 6.43990451466365518223286679974, 7.08820154662815520981181126867, 7.27335930412895876702869607926, 7.75234000453503620370712933227, 7.897885815155730349389946441766, 7.965254244668962494130985274267, 8.740768763493239935399568839108, 8.780491499792404110645782353211, 8.807995506409440880056456620375, 9.900102189037536620764853225375, 9.903128644917207584138040105758, 10.39576802160501566254891546059, 10.63558349084430931817762020080

Graph of the ZZ-function along the critical line